Statistical Methods for Psychology

Similarly, for the sample standard deviation

Recently people whose opinions I respect have suggested that I should remove such

formulae as these from the book because people rarely calculate variances by hand any-

more. Although that is true, and I only wave my hands at most formulae in my own

courses, many people still believe it is important to be able to do the calculation. More im-

portant, perhaps, is the fact that we will see these formulae again in different disguises, and

it helps to understand what is going on if you recognize them for what they are. However,

I agree with those critics in the case of more complex formulae, and in those cases I have

restructured recent editions of the text around definitional formulae.

Applying the computational formula for the sample variance for Set 4, we obtain

Note that the answer we obtained here is exactly the same as the answer we obtained by the

definitional formula. Note also, as pointed out earlier, that is quite differ-

ent from I leave the calculation of the variance for Set 32

to you.

You might be somewhat reassured to learn that the level of mathematics required for

the previous calculations is about as much as you will need anywhere in this book—not

because I am watering down the material, but because an understanding of most applied

statistics does not require much in the way of advanced mathematics. (I told you that you

learned it all in high school.)

The Influence of Extreme Values on the Variance

and Standard Deviation

The variance and standard deviation are very sensitive to extreme scores. To put this differ-

ently, extreme scores play a disproportionate role in determining the variance. Consider a set

of data that range from roughly 0 to 10, with a mean of 5. From the definitional formula for

the variance, you will see that a score of 5 (the mean) contributes nothing to the variance,

because the deviation score is 0. A score of 6 contributes 1/(N 2 1) to , since

A score of 10, however, contributes 25/(N 2 1) units to , since

(10 2 5)^25 25. Thus, although 6 and 10 deviate from the mean by 1 and 5 units, respectively,

their relative contributions to the variance are 1 and 25. This is what we mean when we say

(X 2 X)^2 =(6 2 5)^2 =1. s^2

s^2

(gX)^2 =52.89^2 =2797.35.

gX^2 =148.0241

=

148.0241 2

52.89^2

20

19

=0.4293

=

1.20^21 1.82^2 1 Á 1 4.02^22

52.89^2

20

19

s^2 X=

aX

22 (gX)

2

N

N 21

=

T

aX

221 gX^2

2

N

N 21

sX=

B

a(X^2 X)

2

N 21

Section 2.8 Measures of Variability 43